A practical solution to the shape optimization problem of solid structures

Abstract

This paper presents a practical optimization method for the shape design of solid structures or 3-dimensional structures in order to obtain the optimal free boundary shape without any parameterization of the shape for optimization. A solution to the rigidity design problem of a solid structure under the assumption that the Neumann boundary is allowed to vary is presented. The compliance is minimized subject to a volume constraint and the state equation. Surface tractions, body forces and hydrostatic pressure are applied on the specified regions. This design problem is formulated as a non-parametric shape optimization problem. The shape gradient function is theoretically derived using the Lagrange multiplier method, the material derivative method and the adjoint variable method. With the shape gradient function and the traction method that was proposed by the authors as a gradient method in a Hilbert space, the smooth optimal shape can be easily obtained. This solution is applied to four design problems. The results obtained verified the effectiveness and practical utility of the proposed method for the shape design of solid structures with variable Neumann boundaries. © 2009 WIT Press.